The box is sometimes visible for a friend and sometimes not visible
for another friend. I don't know why.

Why is that?

I'm sorry mate but I think I wasn't able to grasp and fully understood your question.

But just my experience in using fb as I am not really active on being there, I am only using the FB official search bar at the top of the page and type "find friend" and it will give you the page where you would be able to find some of your other friends through your SNS or mail contacts.